MM1A1: Students Will Explore and Interpret the Characteristics of Functions Using Graphs

3.1 Instruction

MM1A1: Students will explore and interpret the characteristics of functions using graphs, tables, and simple algebraic techniques.

MM1A1b: Graph the basic functions f(x) = , where n = 1 to 3, f(x) = , f(x) = |x|, and f(x) = .

MM1A1c: Graph transformations of basic functions including vertical shifts, stretches, and shrinks, as well as reflections across the x- and y-axes.

MM1A1d: Investigate and explain the characteristics of a function: domain, range, zeros, intercepts, intervals of increase and decrease, maximum and minimum values, and end behavior.

MM1A1h: Determine graphically and algebraically whether a function has symmetry and whether it is even, odd, or neither.

I. Cubic Functions

Cubic functions are nonlinear and can be written in the standard form

y = x³ + bx² + cx + d where a ≠ 0.

A function is an odd function if f(-x) = -f(x). The graphs of odd functions are symmetric about the origin.
A function is an even function if f(-x) = f(x). The graphs of even functions are symmetric about the y-axis.
The end behavior of a function’s graph is the behavior of the graph as x approaches positive infinity (+) or negative infinity (-).
If the degree is odd and the leading coefficient is positive: f(x) as x and f(x) as .
If the degree is odd and the leading coefficient is negative: f(x) as x and f(x) as x .

I. Graph y = x³ + c.

Make a table of values for x and y.
Plot points from the table and connect them with a smooth curve.

Example: Graph y = x³ + 3. Compare the graph with the graph of y = x³.

X / -2 / -1 / 0 / 1 / 2
Y / -5 / 2 / 3 / 4 / 11

Students do the following:

y = x³ + 4
y = x³ - 5

II. Graph y = ax³.

Make a table of values for x and y.
Plot points from the table and connect them with a smooth curve.

Example: Graph y = -2x³. Compare the graph with the graph of y = x³.

X / -2 / -1 / 0 / 1 / 2
Y / 16 / 2 / 0 / -2 / -16

Students do the following:

y = 3x³
y = -x³

III. Analyze cubic functions

Determine if the function is even, odd, or neither.
Determine if the graph of the function has symmetry.
A function is an odd function if f(-x) = -f(x). The graphs of odd functions are symmetric about the origin.
A function is an even function if f(-x) = f(x). The graphs of even functions are symmetric about the y-axis.

Determine the intervals of increase and decrease of the graph of the function. (In other words, when does the graph rise and when does it fall?)

Example: Consider the cubic function f(x) = .

Tell whether the function is even, odd, or neither. Does the graph of the function

have symmetry?

Identify the intervals of increase and decrease of the graph of the function.

Students will do the following:

Consider the cubic function f(x) = -x³ - 2x. Answer a and b from the example above for this function.
Consider the cubic function f(x) = -x³ + 2. Answer a and b from the example above for this function.
Consider the cubic function f(x) = -0.4x³ - 2x². Answer a and b from the example above for this function.

3.2 Instruction

MM1A2: Students will simplify and operate with radical expressions, polynomials, and rational expressions.

MM1A2f: Factor expressions by greatest common factor, grouping, trial and error, and special products.

I. Special Products

(x + y)³ = x³ + 3x²y + 3xy² + y³

(x – y)³ = x³ - 3x²y + 3xy² - y³

Examples:

1. a³ + 9a² + 27a + 272. 125 – 75y + 15y² - y³

= (a)³ + 3(a)²(3) + 3(a)(3)² + (3)³ = (5)³ + 3(5)²(-y) + 3(5)(-y)² + (-y)³

= (a + 3)³ = (5 – y)³

Students will do the following:

II. Factor Out a Monomial First

If a cubic has a common monomial factor, factor that first.

Example:

16x – 72xy + 108xy² - 54xy³

= 2x(8 – 36y + 54y² - 27y³)

= 2x[(2)³ - 3(2)²(3y) + 3(2)(3y)² - (3y)³]

= 2x(2 – 3y)³

Students will do the following:

2. k – 6gk + 12g²k - 8g³k

III. Factor Cubics with Multiple Variables

Example:

1. r³t³ + 6r²t² + 12rt + 82. 125x³ - 75x²y + 15xy² - y³

= = (5x)³ - 3(5x)²y + 3(5x)y² - (y)³

= (rt + 2)³ = (5x – y)³

Students will do the following:

54x³a – 108x²a + 72xa – 16a
a³b³ + 18a²b² +108ab + 216

3.3 Instruction

MM1A1: Students will explore and interpret the characteristics of functions using graphs, tables, and simple algebraic techniques.

MM1A1b: Graph the basic functions f(x) = , where n = 1 to 3, f(x) = , f(x) = |x|, and f(x) = .

MM1A1c: Graph transformations of basic functions including vertical shifts, stretches, and shrinks, as well as reflections across the x- and y-axes.

MM1A1d: Investigate and explain the characteristics of a function: domain, range, zeros, intercepts, intervals of increase and decrease, maximum and minimum values, and end behavior.

Vocabulary

A radical expression is an expression that contains a radical, such as a square root, cube root, or other root.

A radical function contains a radical expression with the independent variable in the radicand.

If the radical is a square root, then the function is called a square root function.

The most basic square root function in the family of all square root functions, called the parent square root function, is y = .

I. Graph y = where |a| >1.

Make a table. Because the square root of a negative number is undefined, x must be non-negative. So the domain is x ≥ 0.
Plot the points.
Draw a smooth curve through the points. From either the table or the graph, you can see the range of the function is y ≥ 0.
Compare the graph with the graph of y = .

Example.

1. Graph and identify its domain and range. Compare the graph with the graph of y = .

X / 0 / 1 / 2 / 3
Y / 0 / 2 / 2.8 / 3.5

Students: Graph y = 5. Compare to the graph of y = .

II. Graph y =

Follow the same steps as above. The domain will be x < 0.

Example.

Graph the function y = and identify its domain and range. Compare the graph with the graph of y = .

X / -4 / -3 / -2 / -1 / 0
Y / 1.3 / 1.2 / 0.9 / 0.7 / 0

Students: Graph y = . Compare to the graph of y = .

III. Graph y = where |a| < 1

Follow the same steps as above. The domain will be x > 0.

Example.

Graph the function and identify its domain and range. Compare the graph with the graph of y = .

X / 0 / 1 / 2 / 3 / 4
Y / 0 / -0.25 / -0.35 / -0.43 / -0.5

Students: Graph y = . Compare to the graph of y = .

IV. Graph a Function in the Form y = + k

Example

Graph the function y = + 3 and identify its domain and range. Compare the graph with the graph of y = .

X / 0 / 1 / 2 / 3 / 4
Y / 3 / 4 / 4.4 / 4.7 / 5

Students: Graph y = -2.

V. Graph a function in the form y = a + k

Shift the graph |h| units horizontally and |k| units vertically. Notice that

y = a +k = a + k.|

Example

Graph the function y = .

*First, sketch the graph of y = 2.

X / 0 / 1 / 2 / 3 / 4 / 5 / 6 / 7
Y / 0 / 2 / 2.8 / 3.5 / 4 / 4.5 / 4.9 / 5.3

Students: Graph y = 4 + 2.